Tame and wild automorphisms of polynomials and free algebras

par Prof. Ivan Chestakov (University of São Paulo, Brazil and Sobolev Institute of Mathematics, Novosibirsk, Russia)

jeudi 17 nov. 2016, 14:00 → 15:00 Europe/Paris

125 (bât. Braconnier)

We consider an algebra of polynomials (or a free associative algebra) on n variables (generators) and its group of automorphisms. An automorphism is called elementary if it transforms one of the variables (generators) into a new one by multiplying the old one by a non-zero constant and adding to it an element of the algebra depending only on other variables (generators). The subgroup generated by such automorphisms is called Tame, and its elements are called tame automorphisms. In 1942 Jung proved that, for the case of polynomials in two variablesall automorphisms are tame. In the beginning of 70-s, Makar-Limanov and Czerniakiewicz proved the same result for free associative algebras with two generators. In both cases, it remained an open question whether the same is true for algebras with more than two variables (generators). In 1972 Nagata constructed an automorphism of the algebra of polynomials with three variables A3 which he suggested to be non-tame (wild). Later Anick provided a candidate for a wild automorphism in the free associative algebra on 3 generators. In 2004, Umirbaev and the speaker solved the problem of wild automorphisms in the algebra of polynomials A3 by proving that the Nagata automorphism is wild. Lately, Umirbaev has proved that the Anick automorphism is wild as well. In our talk, we will give some ideas and methods of the proofs of these results and will formulate some new results and conjectures. In particular, we present a wild automorphism in the free Jordan algebra on three generators.